Birth and Death of Invariant Tori for Hamiltonian Systems

The stability of invariant tori with (highly) irrational periods in the context of nearly integrable Hamiltonian systems and symplectic diffeomorphysms is considered. A theorem, based on computer-assisted implementations of recent KAM techniques, establishing the persistence of “golden-mean-tori” for “large” values of the nonlinearity parameter ε in two paradigmatic models is presented. Numerical investigations on the distribution of complex singularities in the parameter e indicates that the method of proof, which involves the explicit construction of approximating surfaces, is optimal at least in the models considered here.